Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa:
https://mathoverflow.net/questions/289414/are-finite-spaces-a-model-for-finite-cw-complexes

So, when it comes to computing homotopy groups of finitely triangulated spaces there is really no loss of generality.

One might draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.